Conditional Spectral Analysis of Replicated Multiple Time Series With Application to Nocturnal Physiology

Abstract

This article considers the problem of analyzing associations between power spectra of multiple time series and cross-sectional outcomes when data are observed from multiple subjects. The motivating application comes from sleep medicine, where researchers are able to non-invasively record physiological time series signals during sleep. The frequency patterns of these signals, which can be quantified through the power spectrum, contain interpretable information about biological processes. An important problem in sleep research is drawing connections between power spectra of time series signals and clinical characteristics; these connections are key to understanding biological pathways through which sleep affects, and can be treated to improve, health. Such analyses are challenging as they must overcome the complicated structure of a power spectrum from multiple time series as a complex positive-definite matrix-valued function. This article proposes a new approach to such analyses based on a tensor-product spline model of Cholesky components of outcome-dependent power spectra. The approach exibly models power spectra as nonparametric functions of frequency and outcome while preserving geometric constraints. Formulated in a fully Bayesian framework, a Whittle likelihood based Markov chain Monte Carlo (MCMC) algorithm is developed for automated model fitting and for conducting inference on associations between outcomes and spectral measures. The method is used to analyze data from a study of sleep in older adults and uncovers new insights into how stress and arousal are connected to the amount of time one spends in bed.

Functional mixed effects wavelet estimation for spectra of replicated time series

“…multivariate EEG time series data from different regions in the brain). Recent unpublished work by [22] treats the problem of analyzing associations between power spectra of multivariate time series and cross-sectional outcomes by an approach based on a tensor-product spline model, in frequency and outcome, of Cholesky components of outcome-dependent power spectra. However, to the best of our knowledge, no quantitative analysis that embeds replicate-specific spectral matrices into a multivariate functional mixed effects model exists so far, not even for the case of independent replicates.…”

Conditional adaptive Bayesian spectral analysis of replicated multivariate time series

“…First, rescaling covariate values to be between 0 and 1 allows for more accurate estimation and inference on the local power spectrum around rescaled covariates by increasing the number of subjects in the sample. Second, our model can be interpreted as the conditional power spectrum, 23 which establishes a connection between the covariate‐dependent power spectrum and the conditional autocovariance matrix. Finally, currently available models 22,23 require the transfer function and covariate‐dependent power spectrum to be continuous in both covariate and frequency.…”

“…In order to preserve the positive‐definiteness of the power spectrum and allow for flexible smoothing among different components of the power spectrum, we model the Cholesky components of the covariate‐dependent power spectrum based on its modified Cholesky decomposition 12,17,23 where is a complex‐valued lower triangular matrix with ones on the diagonal and is a positive diagonal matrix. This leads to the local modified Cholesky decomposition for the ℓ th group for ℓ = 1, … , L .…”

Adaptive Bayesian Time–Frequency Analysis of Multivariate Time Series

“…Since coefficients decay rapidly, using S < L q basis functions presents considerable computational savings without sacrificing model fit. In simulation studies and data analysis, we select S = 10, which accounts for at least 99.975% of the total variance of the full smoothing spline when n q ≤ 10 4 (Krafty et al, 2017).…”

“…A number of Bayesian approaches to spectrum analysis have been explored in the literature. These include methods for analyzing stationary (Carter and Kohn, 1996; Cadonna et al, 2017; Choudhuri et al, 2004; Rosen and Stoffer, 2007; Macaro and Prado, 2014; Krafty et al, 2017) and nonstationary (Rosen et al, 2012; Zhang, 2016; Bruce et al, 2017) time series. Adaptive spectral analysis was introduced by Rosen et al (2012) as a Bayesian approach to univariate nonstationary spectrum analysis, and was latter extended to the multivariate nonstationary setting by Zhang (2016).…”